# Extending the cv package

#### John Fox and Georges Monette

#### 2024-01-23

The cv package is designed to be extensible in several directions. In this vignette, we discuss three kinds of extensions, ordered by increasing general complexity: (1) adding a cross-validation cost criterion; (2) adding a model class that’s not directly accommodated by the `cv()` default method or by another directly inherited method, with separate consideration of mixed-effects models; and (3) adding a new model-selection procedure suitable for use with `selectModel()`.

## Adding a cost criterion

A cost criterion suitable for use with `cv()` or `cvSelect()` should take two arguments, `y` (the observed response vector) and `yhat` (a vector of fitted or predicted response values), and return a numeric index of lack of fit. The cv package supplies several such criteria: `mse(y, yhat)`, which returns the mean-squared prediction error for a numeric response; `rmse(y, yhat)`, which returns the (square-)root mean-squared error; `medAbsErr(y, yhat)`, which returns the median absolute error; and `BayesRule(y, yhat)` (and its non-error-checking version, `BayesRule2(y, yhat))`, suitable for use with a binary regression model, where `y` is the binary response coded `0` for a “failure” or `1` for a “success”; where `yhat` is the predicted probability of success; and where the proportion of incorrectly classified cases is returned.

To illustrate using a different prediction cost criterion, we’ll base a cost criterion on the area under the receiver operating characteristic (“ROC”) curve for a logistic regression. The ROC curve is a graphical representation of the classification power of a binary regression model, and the area under the ROC curve (“AUC”), which varies from 0 to 1, is a common summary measure based on the ROC (see "Receiver operating characteristic", 2023). The Metrics package (Hamner & Frasco, 2018) includes a variety of measures useful for model selection, including an `auc()` function. We convert the AUC into a cost measure by taking its complement:

    AUCcomp <- function(y, yhat) 1 - Metrics::auc(y, yhat)

We then apply `AUCcomp()` to the the Mroz logistic regression discussed in the main cv package vignette, which we reproduce here, using the `Mroz` data frame from the carData package (Fox & Weisberg, 2019):

    data("Mroz", package="carData")
    m.mroz <- glm(lfp ~ ., data=Mroz, family=binomial)
    summary(m.mroz)
    #>
    #> Call:
    #> glm(formula = lfp ~ ., family = binomial, data = Mroz)
    #>
    #> Coefficients:
    #>             Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)  3.18214    0.64438    4.94  7.9e-07 ***
    #> k5          -1.46291    0.19700   -7.43  1.1e-13 ***
    #> k618        -0.06457    0.06800   -0.95  0.34234
    #> age         -0.06287    0.01278   -4.92  8.7e-07 ***
    #> wcyes        0.80727    0.22998    3.51  0.00045 ***
    #> hcyes        0.11173    0.20604    0.54  0.58762
    #> lwg          0.60469    0.15082    4.01  6.1e-05 ***
    #> inc         -0.03445    0.00821   -4.20  2.7e-05 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for binomial family taken to be 1)
    #>
    #>     Null deviance: 1029.75  on 752  degrees of freedom
    #> Residual deviance:  905.27  on 745  degrees of freedom
    #> AIC: 921.3
    #>
    #> Number of Fisher Scoring iterations: 4

    AUCcomp(with(Mroz, as.numeric(lfp == "yes")), fitted(m.mroz))
    #> [1] 0.26362

Cross-validating this cost measure is straightforward:

    library("cv")
    #> Loading required package: doParallel
    #> Loading required package: foreach
    #> Loading required package: iterators
    #> Loading required package: parallel
    cv(m.mroz, criterion=AUCcomp, seed=3639)
    #> R RNG seed set to 3639
    #> 10-Fold Cross Validation
    #> criterion: AUCcomp
    #> cross-validation criterion = 0.27471
    #> full-sample criterion = 0.26362

As expected, the cross-validated complement to the AUC is somewhat less optimistic than the criterion computed from the model fit to the whole data set.

As we explain in the vignette “Cross-validation of regression models,” the `cv()` function differentiates between CV criteria that are averages of casewise components and criteria that are not. Computation of bias corrections and confidence intervals is limited to the former. We show in the appendix to this vignette that the AUC, and hence its complement, cannot be expressed as averages of casewise components.

`cv()` looks for a `"casewise loss"` attribute of the value returned by a CV criterion function. If this attribute exists, then the criterion is treated as the mean of casewise components, and `cv()` uses the unexported function `getLossFn()` to construct a function that returns the casewise components of the criterion.

We illustrate with the `mse()`:

    mse
    #> function (y, yhat)
    #> {
    #>     result <- mean((y - yhat)^2)
    #>     attr(result, "casewise loss") <- "(y - yhat)^2"
    #>     result
    #> }
    #> <bytecode: 0x10dacf2f8>
    #> <environment: namespace:cv>

    cv:::getLossFn(mse(rnorm(100), rnorm(100)))
    #> function (y, yhat)
    #> {
    #>     (y - yhat)^2
    #> }
    #> <environment: 0x10db33828>

For this scheme to work, the “casewise loss” attribute must be a character string (or vector of character strings), here `"(y - yhat)^2"`, that evaluates to an expression that is a function of `y` and `yhat`, and that computes the vector of casewise components of the CV criterion.

## Adding a model class not covered by the default `cv()` method

### Independently sampled cases

Suppose that we want to cross-validate a multinomial logistic regression model fit by the `multinom()` function in the nnet package (Venables & Ripley, 2002). We borrow an example from Fox (2016, sec. 14.2.1), with data from the British Election Panel Study on vote choice in the 2001 British election. Data for the example are in the `BEPS` data frame in the carData package:

    data("BEPS", package="carData")
    head(BEPS)
    #>               vote age economic.cond.national economic.cond.household Blair
    #> 1 Liberal Democrat  43                      3                       3     4
    #> 2           Labour  36                      4                       4     4
    #> 3           Labour  35                      4                       4     5
    #> 4           Labour  24                      4                       2     2
    #> 5           Labour  41                      2                       2     1
    #> 6           Labour  47                      3                       4     4
    #>   Hague Kennedy Europe political.knowledge gender
    #> 1     1       4      2                   2 female
    #> 2     4       4      5                   2   male
    #> 3     2       3      3                   2   male
    #> 4     1       3      4                   0 female
    #> 5     1       4      6                   2   male
    #> 6     4       2      4                   2   male

The polytomous (multi-category) response variable is `vote`, a factor with levels `"Conservative"`, `"Labour"`, and `"Liberal Democrat"`. The predictors of `vote` are:

- `age`, in years;
- `econ.cond.national` and `econ.cond.household`, the respondent’s ratings of the state of the economy, on 1 to 5 scales.
- `Blair`, `Hague`, and `Kennedy`, ratings of the leaders of the Labour, Conservative, and Liberal Democratic parties, on 1 to 5 scales.
- `Europe`, an 11-point scale on attitude towards European integration, with high scores representing “Euro-skepticism.”
- `political.knowledge`, knowledge of the parties’ positions on European integration, with scores from 0 to 3.
- `gender`, `"female"` or `"male"`.

The model fit to the data includes an interaction between `Europe` and `political.knowledge`; the other predictors enter the model additively:

    library("nnet")
    m.beps <- multinom(vote ~ age + gender + economic.cond.national +
                           economic.cond.household + Blair + Hague + Kennedy +
                           Europe*political.knowledge, data=BEPS)
    #> # weights:  36 (22 variable)
    #> initial  value 1675.383740
    #> iter  10 value 1240.047788
    #> iter  20 value 1163.199642
    #> iter  30 value 1116.519687
    #> final  value 1116.519666
    #> converged

    car::Anova(m.beps)
    #> Analysis of Deviance Table (Type II tests)
    #>
    #> Response: vote
    #>                            LR Chisq Df Pr(>Chisq)
    #> age                            13.9  2    0.00097 ***
    #> gender                          0.5  2    0.79726
    #> economic.cond.national         30.6  2    2.3e-07 ***
    #> economic.cond.household         5.7  2    0.05926 .
    #> Blair                         135.4  2    < 2e-16 ***
    #> Hague                         166.8  2    < 2e-16 ***
    #> Kennedy                        68.9  2    1.1e-15 ***
    #> Europe                         78.0  2    < 2e-16 ***
    #> political.knowledge            55.6  2    8.6e-13 ***
    #> Europe:political.knowledge     50.8  2    9.3e-12 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Most of the predictors, including the `Europe` \(\times\) `political.knowledge` interaction, are associated with very small \(p\)-values; the `Anova()` function is from the car package (Fox & Weisberg, 2019).

Here’s an “effect plot”, using the the effects package (Fox & Weisberg, 2019) to visualize the `Europe` \(\times\) `political.knowledge` interaction in a “stacked-area” graph:

    plot(effects::Effect(c("Europe", "political.knowledge"), m.beps,
                xlevels=list(Europe=1:11, political.knowledge=0:3),
                fixed.predictors=list(given.values=c(gendermale=0.5))),
         lines=list(col=c("blue", "red", "orange")),
         axes=list(x=list(rug=FALSE), y=list(style="stacked")))

To cross-validate this multinomial-logit model we need an appropriate cost criterion. None of the criteria supplied by the cv package—for example, neither `mse()`, which is appropriate for a numeric response, nor `BayesRule()`, which is appropriate for a binary response—will do. One possibility is to adapt Bayes rule to a polytomous response:

    head(BEPS$vote)
    #> [1] Liberal Democrat Labour           Labour           Labour
    #> [5] Labour           Labour
    #> Levels: Conservative Labour Liberal Democrat
    yhat <- predict(m.beps, type="class")
    head(yhat)
    #> [1] Labour           Labour           Labour           Labour
    #> [5] Liberal Democrat Labour
    #> Levels: Conservative Labour Liberal Democrat

    BayesRuleMulti <- function(y, yhat){
      result <- mean(y != yhat)
      attr(result, "casewise loss") <- "y != yhat"
      result
    }

    BayesRuleMulti(BEPS$vote, yhat)
    #> [1] 0.31869
    #> attr(,"casewise loss")
    #> [1] "y != yhat"

The `predict()` method for `"multinom"` models called with argument `type="class"` reports the Bayes-rule prediction for each case—that is, the response category with the highest predicted probability. Our `BayesRuleMulti()` function calculates the proportion of misclassified cases. Because this value is the mean of casewise components, we attach a `"casewise loss"` attribute to the result (as explained in the preceding section).

The marginal proportions for the response categories are

    xtabs(~ vote, data=BEPS)/nrow(BEPS)
    #> vote
    #>     Conservative           Labour Liberal Democrat
    #>          0.30295          0.47213          0.22492

and so the marginal Bayes-rule prediction, that everyone will vote Labour, produces an error rate of \(1 - 0.47213 = 0.52787\). The multinomial-logit model appears to do substantially better than that, but does its performance hold up to cross-validation?

We check first whether the default `cv()` method works “out-of-the-box” for the `"multinom"` model:

    cv(m.beps, seed=3465, criterion=BayesRuleMulti)
    #> Error in GetResponse.default(model): non-vector response

The default method of `GetResponse()` (a function supplied by the cv package—see `?GetResponse`) fails for a `"multinom"` object. A straightforward solution is to supply a `GetResponse.multinom()` method that returns the factor response (using the `get_response()` function from the insight package, Lüdecke, Waggoner, & Makowski, 2019),

    GetResponse.multinom <- function(model, ...) {
      insight::get_response(model)
    }

    head(GetResponse(m.beps))
    #> [1] Liberal Democrat Labour           Labour           Labour
    #> [5] Labour           Labour
    #> Levels: Conservative Labour Liberal Democrat

and to try again:

    cv(m.beps, seed=3465, criterion=BayesRuleMulti)
    #> R RNG seed set to 3465
    #> # weights:  36 (22 variable)
    #> initial  value 1507.296060
    #> iter  10 value 1134.575036
    #> iter  20 value 1037.413231
    #> iter  30 value 1007.705242
    #> iter  30 value 1007.705235
    #> iter  30 value 1007.705235
    #> final  value 1007.705235
    #> converged
    #> Error in match.arg(type): 'arg' should be one of "class", "probs"

A `traceback()` (not shown) reveals that the problem is that the default method of `cv()` calls the `"multinom"` method for `predict()` with the argument `type="response"`, when the correct argument should be `type="class"`. We therefore must write a “`multinom`” method for `cv()`, but that proves to be very simple:

    cv.multinom <- function (model, data, criterion=BayesRuleMulti, k, reps,
                             seed, ...){
      NextMethod(type="class", criterion=criterion)
    }

That is, we simply call the default `cv()` method with the `type` argument properly set. In addition to supplying the correct `type` argument, our method sets the default `criterion` for the `cv.multinom()` method to `BayesRuleMulti`.

Then:

    m.beps <- update(m.beps, trace=FALSE)
    cv(m.beps, seed=3465)
    #> R RNG seed set to 3465
    #> 10-Fold Cross Validation
    #> cross-validation criterion = 0.32459
    #> bias-adjusted cross-validation criterion = 0.32368
    #> 95% CI for bias-adjusted CV criterion = (0.30017, 0.34718)
    #> full-sample criterion = 0.31869

Prior to invoking `cv()`, we called `update()` with `trace=FALSE` to suppress the iteration history reported by default by `multinom()`—it would be tedious to see the iteration history for each fold. The cross-validated polytomous Bayes-rule criterion confirms that the fitted model does substantially better than the marginal Bayes-rule prediction that everyone votes for Labour.

### Mixed-effects models

Adding a `cv()` method for a mixed-model class is somewhat more complicated. We provide the `cvMixed()` function to facilitate this process, and to see how that works, consider the `"lme"` method from the cv package:

    cv:::cv.lme
    #> function (model, data = insight::get_data(model), criterion = mse,
    #>     k, reps = 1, seed, ncores = 1, clusterVariables, ...)
    #> {
    #>     cvMixed(model, package = "nlme", data = data, criterion = criterion,
    #>         k = k, reps = reps, seed = seed, ncores = ncores, clusterVariables = clusterVariables,
    #>         predict.clusters.args = list(object = model, newdata = data,
    #>             level = 0), predict.cases.args = list(object = model,
    #>             newdata = data, level = 1), ...)
    #> }
    #> <bytecode: 0x11a28c118>
    #> <environment: namespace:cv>

Notice that `cv.lme()` sets up a call to `cvMixed()`, which does the computational work.

Most of the arguments of `cvMixed()` are familiar:

- `model` is the mixed-model object, here of class `"lme"`.

- `package` is the name of the package in which the mixed-modeling function used to fit the model, here `lme()`, resides—i.e., `"nlme"`; `cvMixed()` uses this argument to retrieve the package namespace.

- `data` is the data set to which the model is fit, by default extracted by the `get_data()` function in the insight package.

- `criterion` is the CV criterion, defaulting to the `mse()` function.

- `k` is the number of CV folds, defaulting to `"loo"` for CV by clusters and `10` for CV by cases.

- `reps` is the number of times the CV process is repeated, defaulting to `1`.

- `seed` is the seed for R’s random-number generator, defaulting to a randomly selected (and saved) value.

- `ncores` is the number of cores to use for parallel computation; if `1`, the default, then the computation isn’t parallelized.

- `clusterVariables` is a character vector of the names of variables defining clusters; if missing, then CV is based on cases rather than clusters.

The remaining two arguments are unfamiliar:

- `predict.clusters.args` is a named list of arguments to be passed to the `predict()` function to obtain predictions for the full data set from a model fit to a subset of the data for cluster-based CV. The first two arguments should be `object` and `newdata`. It is typically necessary to tell `cvMixed()` how to base predictions only on fixed effects; in the case of `"lme"` models, this is done by setting `level = 0`.

- Similarly, `predict.cases.args` is a named list of arguments to be passed to `predict()` for case-based CV. Setting `level = 1` includes random effects in the predictions.

Finally, any additional arguments, absorbed by `...`, are passed to `update()` when the model is refit with each fold omitted. `cvMixed()` returns an object of class `"cv"`.

Now imagine that we want to support a new class of mixed-effects models. To be concrete, we illustrate with the `glmmPQL()` function in the MASS package (Venables & Ripley, 2002), which fits generalized-linear mixed-effects models by penalized quasi-likelihood.1 Not coincidentally, the arguments of `glmmPQL()` are similar to those of `lme()` (with an additional `family` argument), because the former iteratively invokes the latter; so `cv.glmmPQL()` should resemble `cv.lme()`.

As it turns out, neither the default method for `GetResponse()` nor `insight::get_data()` work for `"glmmPQL"` objects. These objects include a `"data"` element, however, and so we can simply extract this element as the default for the `data` argument of our `cv.glmmPQL()` method.

To get the response variable is more complicated: We refit the fixed part of the model as a GLM with only the regression constant on the right-hand side, and extract the response from that; because all we need is the response variable, we limit the number of GLM iterations to 1 and suppress warning messages about non-convergence:

    GetResponse.glmmPQL <- function(model, ...){
      f <- formula(model)
      f[[3]] <- 1 # regression constant only on RHS
      model <- suppressWarnings(glm(f, data=model$data, family=model$family,
                                    control=list(maxit=1)))
      cv::GetResponse(model)
    }

Writing the `cv()` method is then straightforward:

    cv.glmmPQL <- function(model, data = model$data, criterion = mse,
                         k, reps = 1, seed, ncores = 1, clusterVariables, ...){
      cvMixed(
        model,
        package="MASS",
        data=data,
        criterion=criterion,
        k=k,
        reps=reps,
        seed=seed,
        ncores=ncores,
        clusterVariables=clusterVariables,
        predict.clusters.args=list(object=model,
                                   newdata=data,
                                   level=0,
                                   type="response"),
        predict.cases.args=list(object=model,
                                newdata=data,
                                level=1,
                                type="response"),
        verbose=FALSE,
        ...)
    }

We set the argument `verbose=FALSE` to suppress `glmmPQL()`’s iteration counter when `cvMixed()` calls `update()`.

Let’s apply our newly minted method to a logistic regression with a random intercept in an example that appears in `?glmmPQL`:

    library("MASS")
    m.pql <- glmmPQL(y ~ trt + I(week > 2), random = ~ 1 | ID,
                 family = binomial, data = bacteria)
    #> iteration 1
    #> iteration 2
    #> iteration 3
    #> iteration 4
    #> iteration 5
    #> iteration 6
    summary(m.pql)
    #> Linear mixed-effects model fit by maximum likelihood
    #>   Data: bacteria
    #>   AIC BIC logLik
    #>    NA  NA     NA
    #>
    #> Random effects:
    #>  Formula: ~1 | ID
    #>         (Intercept) Residual
    #> StdDev:      1.4106  0.78005
    #>
    #> Variance function:
    #>  Structure: fixed weights
    #>  Formula: ~invwt
    #> Fixed effects:  y ~ trt + I(week > 2)
    #>                   Value Std.Error  DF t-value p-value
    #> (Intercept)      3.4120   0.51850 169  6.5805  0.0000
    #> trtdrug         -1.2474   0.64406  47 -1.9367  0.0588
    #> trtdrug+        -0.7543   0.64540  47 -1.1688  0.2484
    #> I(week > 2)TRUE -1.6073   0.35834 169 -4.4853  0.0000
    #>  Correlation:
    #>                 (Intr) trtdrg trtdr+
    #> trtdrug         -0.598
    #> trtdrug+        -0.571  0.460
    #> I(week > 2)TRUE -0.537  0.047 -0.001
    #>
    #> Standardized Within-Group Residuals:
    #>      Min       Q1      Med       Q3      Max
    #> -5.19854  0.15723  0.35131  0.49495  1.74488
    #>
    #> Number of Observations: 220
    #> Number of Groups: 50

We compare this result to that obtained from `glmer()` in the lme4 package:

    library("lme4")
    #> Loading required package: Matrix
    m.glmer <- glmer(y ~ trt + I(week > 2) + (1 | ID),
                   family = binomial, data = bacteria)
    summary(m.glmer)
    #> Generalized linear mixed model fit by maximum likelihood (Laplace
    #>   Approximation) [glmerMod]
    #>  Family: binomial  ( logit )
    #> Formula: y ~ trt + I(week > 2) + (1 | ID)
    #>    Data: bacteria
    #>
    #>      AIC      BIC   logLik deviance df.resid
    #>    202.3    219.2    -96.1    192.3      215
    #>
    #> Scaled residuals:
    #>    Min     1Q Median     3Q    Max
    #> -4.561  0.136  0.302  0.422  1.128
    #>
    #> Random effects:
    #>  Groups Name        Variance Std.Dev.
    #>  ID     (Intercept) 1.54     1.24
    #> Number of obs: 220, groups:  ID, 50
    #>
    #> Fixed effects:
    #>                 Estimate Std. Error z value Pr(>|z|)
    #> (Intercept)        3.548      0.696    5.10  3.4e-07 ***
    #> trtdrug           -1.367      0.677   -2.02  0.04352 *
    #> trtdrug+          -0.783      0.683   -1.15  0.25193
    #> I(week > 2)TRUE   -1.598      0.476   -3.36  0.00078 ***
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Correlation of Fixed Effects:
    #>             (Intr) trtdrg trtdr+
    #> trtdrug     -0.593
    #> trtdrug+    -0.537  0.487
    #> I(wk>2)TRUE -0.656  0.126  0.064

      # comparison of fixed effects:
    car::compareCoefs(m.pql, m.glmer)
    #> Warning in car::compareCoefs(m.pql, m.glmer): models to be compared are of
    #> different classes
    #> Calls:
    #> 1: glmmPQL(fixed = y ~ trt + I(week > 2), random = ~1 | ID, family =
    #>   binomial, data = bacteria)
    #> 2: glmer(formula = y ~ trt + I(week > 2) + (1 | ID), data = bacteria,
    #>   family = binomial)
    #>
    #>                 Model 1 Model 2
    #> (Intercept)       3.412   3.548
    #> SE                0.514   0.696
    #>
    #> trtdrug          -1.247  -1.367
    #> SE                0.638   0.677
    #>
    #> trtdrug+         -0.754  -0.783
    #> SE                0.640   0.683
    #>
    #> I(week > 2)TRUE  -1.607  -1.598
    #> SE                0.355   0.476
    #>

The two sets of estimates are similar, but not identical

Finally, we try out our `cv.glmmPQL()` method, cross-validating both by clusters and by cases,

    cv(m.pql, clusterVariables="ID", criterion=BayesRule)
    #> n-Fold Cross Validation based on 50 {ID} clusters
    #> cross-validation criterion = 0.19545
    #> bias-adjusted cross-validation criterion = 0.19545
    #> full-sample criterion = 0.19545

    cv(m.pql, data=bacteria, criterion=BayesRule, seed=1490)
    #> R RNG seed set to 1490
    #> 10-Fold Cross Validation
    #> cross-validation criterion = 0.20909
    #> bias-adjusted cross-validation criterion = 0.20727
    #> full-sample criterion = 0.14545

and again compare to `glmer()`:

    cv(m.glmer, clusterVariables="ID", criterion=BayesRule)
    #> n-Fold Cross Validation based on 50 {ID} clusters
    #> cross-validation criterion = 0.19545
    #> bias-adjusted cross-validation criterion = 0.19545
    #> full-sample criterion = 0.19545

    cv(m.glmer, data=bacteria, criterion=BayesRule, seed=1490)
    #> R RNG seed set to 1490
    #> 10-Fold Cross Validation
    #> cross-validation criterion = 0.19545
    #> bias-adjusted cross-validation criterion = 0.19364
    #> full-sample criterion = 0.15

## Adding a model-selection procedure

The `selectStepAIC()` function supplied by the cv package, which is based on the `stepAIC()` function from the nnet package (Venables & Ripley, 2002) for stepwise model selection, is suitable for the `procedure` argument of `cvSelect()`. The use of `selectStepAIC()` is illustrated in the principal vignette for the package.

We’ll employ `selectStepAIC()` as a “template” for writing a CV model-selection procedure. To see the code for this function, type `cv::selectStepAIC` at the R command prompt, or examine the sources for the cv package at https://github.com/gmonette/cv (the code for `selectStepAIC()` is in https://github.com/gmonette/cv/blob/main/R/cvSelect.R).

Another approach to model selection is all-subsets regression. The `regsubsets()` function in the leaps package (Lumley & Miller, 2020) implements an efficient algorithm for selecting the best-fitting linear least-squares regressions for subsets of predictors of all sizes, from 1 through the maximum number of candidate predictors.2 To illustrate the use of `regsubsets()`, we employ the `swiss` data frame supplied by the leaps package:

    library("leaps")
    head(swiss)
    #>              Fertility Agriculture Examination Education Catholic
    #> Courtelary        80.2        17.0          15        12     9.96
    #> Delemont          83.1        45.1           6         9    84.84
    #> Franches-Mnt      92.5        39.7           5         5    93.40
    #> Moutier           85.8        36.5          12         7    33.77
    #> Neuveville        76.9        43.5          17        15     5.16
    #> Porrentruy        76.1        35.3           9         7    90.57
    #>              Infant.Mortality
    #> Courtelary               22.2
    #> Delemont                 22.2
    #> Franches-Mnt             20.2
    #> Moutier                  20.3
    #> Neuveville               20.6
    #> Porrentruy               26.6
    nrow(swiss)
    #> [1] 47

The data set includes the following variables, for each of 47 French-speaking Swiss provinces circa 1888:

- `Fertility`: A standardized fertility measure.
- `Agriculture`: The percentage of the male population engaged in agriculture.
- `Examination`: The percentage of draftees into the Swiss army receiving the highest grade on an examination.
- `Education`: The percentage of draftees with more than a primary-school education.
- `Catholic`: The percentage of the population who were Catholic.
- `Infant.Mortality`: The infant-mortality rate, expressed as the percentage of live births surviving less than a year.

Following Lumley & Miller (2020), we treat `Fertility` as the response and the other variables as predictors in a linear least-squares regression:

    m.swiss <- lm(Fertility ~ ., data=swiss)
    summary(m.swiss)
    #>
    #> Call:
    #> lm(formula = Fertility ~ ., data = swiss)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -15.274  -5.262   0.503   4.120  15.321
    #>
    #> Coefficients:
    #>                  Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)       66.9152    10.7060    6.25  1.9e-07 ***
    #> Agriculture       -0.1721     0.0703   -2.45   0.0187 *
    #> Examination       -0.2580     0.2539   -1.02   0.3155
    #> Education         -0.8709     0.1830   -4.76  2.4e-05 ***
    #> Catholic           0.1041     0.0353    2.95   0.0052 **
    #> Infant.Mortality   1.0770     0.3817    2.82   0.0073 **
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 7.17 on 41 degrees of freedom
    #> Multiple R-squared:  0.707,  Adjusted R-squared:  0.671
    #> F-statistic: 19.8 on 5 and 41 DF,  p-value: 5.59e-10

    cv(m.swiss, seed=8433)
    #> R RNG seed set to 8433
    #> 10-Fold Cross Validation
    #> method: Woodbury
    #> criterion: mse
    #> cross-validation criterion = 59.683
    #> bias-adjusted cross-validation criterion = 58.846
    #> full-sample criterion = 44.788

Thus, the RMSE for the model fit to the complete data is considerably smaller than the CV estimate of the RMSE. Can we do better by selecting a subset of the predictors, taking account of the additional uncertainty induced by model selection?

First, let’s apply best-subset selection to the complete data set:

    swiss.sub <- regsubsets(Fertility ~ ., data=swiss)
    summary(swiss.sub)
    #> Subset selection object
    #> Call: regsubsets.formula(Fertility ~ ., data = swiss)
    #> 5 Variables  (and intercept)
    #>                  Forced in Forced out
    #> Agriculture          FALSE      FALSE
    #> Examination          FALSE      FALSE
    #> Education            FALSE      FALSE
    #> Catholic             FALSE      FALSE
    #> Infant.Mortality     FALSE      FALSE
    #> 1 subsets of each size up to 5
    #> Selection Algorithm: exhaustive
    #>          Agriculture Examination Education Catholic Infant.Mortality
    #> 1  ( 1 ) " "         " "         "*"       " "      " "
    #> 2  ( 1 ) " "         " "         "*"       "*"      " "
    #> 3  ( 1 ) " "         " "         "*"       "*"      "*"
    #> 4  ( 1 ) "*"         " "         "*"       "*"      "*"
    #> 5  ( 1 ) "*"         "*"         "*"       "*"      "*"

    (bics <- summary(swiss.sub)$bic)
    #> [1] -19.603 -28.611 -35.656 -37.234 -34.553
    which.min(bics)
    #> [1] 4

    car::subsets(swiss.sub, legend="topright")

Selecting the best model of each size.

The graph, produced by the `subsets()` function in the car package, shows that the model with the smallest BIC is the best model with 4 predictors, including `Agriculture`, `Education`, `Catholic`, and `Infant.Mortality`, but not `Examination`:

    m.best <- update(m.swiss, . ~ . - Examination)
    summary(m.best)
    #>
    #> Call:
    #> lm(formula = Fertility ~ Agriculture + Education + Catholic +
    #>     Infant.Mortality, data = swiss)
    #>
    #> Residuals:
    #>     Min      1Q  Median      3Q     Max
    #> -14.676  -6.052   0.751   3.166  16.142
    #>
    #> Coefficients:
    #>                  Estimate Std. Error t value Pr(>|t|)
    #> (Intercept)       62.1013     9.6049    6.47  8.5e-08 ***
    #> Agriculture       -0.1546     0.0682   -2.27   0.0286 *
    #> Education         -0.9803     0.1481   -6.62  5.1e-08 ***
    #> Catholic           0.1247     0.0289    4.31  9.5e-05 ***
    #> Infant.Mortality   1.0784     0.3819    2.82   0.0072 **
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> Residual standard error: 7.17 on 42 degrees of freedom
    #> Multiple R-squared:  0.699,  Adjusted R-squared:  0.671
    #> F-statistic: 24.4 on 4 and 42 DF,  p-value: 1.72e-10

    cv(m.best, seed=8433) # use same folds as before
    #> R RNG seed set to 8433
    #> 10-Fold Cross Validation
    #> method: Woodbury
    #> criterion: mse
    #> cross-validation criterion = 58.467
    #> bias-adjusted cross-validation criterion = 57.778
    #> full-sample criterion = 45.916

The RMSE for the selected model is (of course) slightly higher than for the full model fit previously, but the cross-validated RMSE is a bit lower; as we explain in the main vignette, however, it isn’t kosher to select and cross-validate a model on the same data.

Here’s a function named `selectSubsets()`, meant to be used with `cvSelect()`, suitable for cross-validating the model-selection process:

    selectSubsets <- function(data=insight::get_data(model),
                              model,
                              indices,
                              criterion=mse,
                              save.coef=TRUE, ...){

      if (inherits(model, "lm", which=TRUE) != 1)
        stop("selectSubsets is appropriate only for 'lm' models")

      y <- GetResponse(model)
      formula <- formula(model)
      X <- model.matrix(model)

      if (missing(indices)) {
        # select the best model from the full data by BIC
        sel <- leaps::regsubsets(formula, data=data, ...)
        bics <- summary(sel)$bic
        best <- coef(sel, 1:length(bics))[[which.min(bics)]]
        x.names <- names(best)
        # fit the best model; intercept is already in X, hence - 1:
        m.best <- lm(y ~ X[, x.names] - 1)
        fit.all <- predict(m.best, newdata=data)
        return(criterion(y, fit.all)) # return the CV criterion
      }

      # select the best model omitting the i-th fold (given by indices)
      sel.i <- leaps::regsubsets(formula, data[-indices, ], ...)
      bics.i <- summary(sel.i)$bic
      best.i <- coef(sel.i, 1:length(bics.i))[[which.min(bics.i)]]
      x.names.i <- names(best.i)
      m.best.i <- lm(y[-indices] ~ X[-indices, x.names.i] - 1)
                  # predict() doesn't work here:
      fit.all.i <- as.vector(X[, x.names.i] %*% coef(m.best.i))
      fit.i <- fit.all.i[indices]
      # return the fitted values for i-th fold, CV criterion for all cases,
      #   and the regression coefficients
      list(fit.i=fit.i, # fitted values for i-th fold
           crit.all.i=criterion(y, fit.all.i), # CV crit for all cases
           coefficients = if (save.coef){ # regression coefficients
             coefs <- coef(m.best.i)

             # fix coefficient names
             names(coefs) <- sub("X\\[-indices, x.names.i\\]", "",
                                 names(coefs))

             coefs
           }  else {
             NULL
           }
      )
    }

A slightly tricky point is that because of scoping issues, `predict()` doesn’t work with the model fit omitting the \(i\)th fold, and so the fitted values for all cases are computed directly as \(\widehat{\mathbf{y}}_{-i} = \mathbf{X} \mathbf{b}_{-i}\), where \(\mathbf{X}\) is the model-matrix for all of the cases, and \(\mathbf{b}_{-i}\) is the vector of least-squares coefficients for the selected model with the \(i\)th fold omitted.

Additionally, the command `lm(y[-indices] ~ X[-indices, x.names.i] - 1)`, which is the selected model with the \(i\)th fold deleted, produces awkward coefficient names like `"X[-indices, x.names.i]Infant.Mortality"`. Purely for aesthetic reasons, the command `sub("X\\[-indices, x.names.i\\]", "", names(coefs))` fixes these awkward names, removing the extraneous text, `"X[-indices, x.names.i]"`.

Applying `selectSubsets()` to the full data produces the full-data cross-validated RMSE (which we obtained previously):

    selectSubsets(model=m.swiss)
    #> [1] 45.916
    #> attr(,"casewise loss")
    #> [1] "(y - yhat)^2"

Similarly, applying the function to an imaginary “fold” of 5 cases returns the RMSE for the cases in the fold, based on the model selected and fit to the cases omitting the fold; the RMSE for all of the cases, based on the same model; and the coefficients of the selected model, which includes 4 or the 5 predictors (and the intercept):

    selectSubsets(model=m.swiss, indices=seq(5, 45, by=10))
    #> $fit.i
    #> [1] 62.922 67.001 73.157 83.778 32.251
    #>
    #> $crit.all.i
    #> [1] 46.297
    #> attr(,"casewise loss")
    #> [1] "(y - yhat)^2"
    #>
    #> $coefficients
    #>      (Intercept)      Agriculture        Education         Catholic
    #>         63.80452         -0.15895         -1.04218          0.13066
    #> Infant.Mortality
    #>          1.01895

Then, using `selectSubsets()` in cross-validation, we get:

    (cv.swiss <- cvSelect(selectSubsets, model=m.swiss,
                          data=swiss, seed=8433)) # use same folds
    #> R RNG seed set to 8433
    #> 10-Fold Cross Validation
    #> cross-validation criterion = 65.835
    #> bias-adjusted cross-validation criterion = 63.644
    #> full-sample criterion = 45.916

Cross-validation shows that model selection exacts a penalty in RMSE. Examining the models selected for the 10 folds reveals that there is some uncertainty in identifying the predictors in the “best” model, with `Agriculture` sometimes appearing and sometimes not:

    compareFolds(cv.swiss)
    #>         (Intercept) Catholic Education Infant.Mortality Agriculture
    #> Fold 1      59.0852   0.1397   -1.0203           1.2985       -0.17
    #> Fold 2      67.0335   0.1367   -1.0499           0.9413       -0.20
    #> Fold 3      55.0453   0.1221   -0.8757           1.3541       -0.15
    #> Fold 4      62.5543   0.1236   -0.9719           1.0679       -0.16
    #> Fold 5      50.4643   0.1057   -0.7863           1.2144
    #> Fold 6      68.0289   0.1195   -1.0073           0.8294       -0.17
    #> Fold 7      66.5219   0.1357   -1.0827           0.9523       -0.19
    #> Fold 8      46.3507   0.0776   -0.7637           1.4463
    #> Fold 9      62.2632   0.1230   -1.0067           1.1000       -0.17
    #> Fold 10     52.5112   0.1005   -0.7232           1.0809

## Appendix: Why the complement of AUC isn’t a casewise CV criterion

Consider calculating AUC for folds in which a validation set contains \(n_v\) observations. To calculate AUC in the validation set, we need the vector of prediction criteria, \(\widehat{\mathbf{y}}_{v_{(n_v \times 1)}} = (\widehat{y}_1, ..., \widehat{y}_{n_v})^T\), and the vector of observed responses in the validation set, \(\mathbf{y}_{v_{(n_v \times 1)}} = (y_1, \ldots, y_{n_v})^T\) with \(y_i \in \{0,1\}, \; i = 1, \ldots, n_v\).

To construct the ROC curve, only the ordering of the values in \(\mathbf{\widehat{y}}_v\) is relevant. Thus, assuming that there are no ties, and reordering observations if necessary, we can set \(\mathbf{\widehat{y}}_v = (1, 2, \ldots, n_v)^T\).

If the AUC can be expressed as the casewise mean or sum of a function \(\mathrm{cv}(\widehat{y}_i,y_i)\), where \(\mathrm{cv}: \{1,2,...,n_v\}\times\{0,1\} \rightarrow [0,1]\), then \[\begin{equation} \label{eq:cw} \tag{1} \sum_{i=1}^{n_v} \mathrm{cv}(\widehat{y}_i,y_i) = \mathrm{AUC}(\mathbf{\widehat{y}}_v,\mathbf{y}_v) \end{equation}\] must hold for all \(2^{n_v}\) possible values of \(\mathbf{y}_v = (y_1,...,y_{n_v})^T\). If all \(y\mathrm{s}\) have the same value, either 1 or 0, then the definition of AUC is ambiguous. AUC could be considered undefined, or it could be set to 0 if all \(y\)s are 0 and to 1 if all \(y\)s are 1. If AUC is considered to be undefined in these cases, we have \(2^{n_v} - 2\) admissible values for \(\mathbf{y}_v\).

Thus, equation (\(\ref{eq:cw}\)) produces either \(2^{n_v}\) or \(2^{n_v}-2\) constraints. Although there are only \(2n_v\) possible values for the \(\mathrm{cv(\cdot)}\) function, equation (\(\ref{eq:cw}\)) could, nevertheless, have consistent solutions. We therefore need to determine whether there is a value of \(n_v\) for which (\(\ref{eq:cw}\)) has no consistent solution for all admissible values of \(\mathbf{y}_v\). In that eventuality, we will have shown that AUC cannot, in general, be expressed through a casewise sum.

If \(n_v=3\), we show below that (\(\ref{eq:cw}\)) has no consistent solution if we include all possibilities for \(\mathbf{y}_v\), but does if we exclude cases where all \(y\)s have the same value. If \(n_v=4\), we show that there are no consistent solutions in either case.

The following R function computes AUC from \(\mathbf{\widehat{y}}_v\) and \(\mathbf{y}_v\), accommodating the cases where \(\mathbf{y}_v\) is all 0s or all 1s:

    AUC <- function(y, yhat = seq_along(y)) {
      s <- sum(y)
      if (s == 0) return(0)
      if (s == length(y)) return(1)
      Metrics::auc(y, yhat)
    }

We then define a function to generate all possible \(\mathbf{y}_v\)s of length \(n_v\) as rows of the matrix \(\mathbf{Y}_{(2^{n_v} \times n_v)}\):

    Ymat <- function(n_v, exclude_identical = FALSE) {
      stopifnot(n_v > 0 && round(n_v) == n_v)    # n_v must be a positive integer
      ret <- sapply(0:(2^n_v - 1),
                    function(x) as.integer(intToBits(x)) )[1:n_v, ]
      ret <- if (is.matrix(ret)) t(ret) else matrix(ret)
      colnames(ret) <- paste0("y", 1:ncol(ret))
      if (exclude_identical) ret[-c(1, nrow(ret)), ] else ret
    }

For \(n_v=3\),

    Ymat(3)
    #>      y1 y2 y3
    #> [1,]  0  0  0
    #> [2,]  1  0  0
    #> [3,]  0  1  0
    #> [4,]  1  1  0
    #> [5,]  0  0  1
    #> [6,]  1  0  1
    #> [7,]  0  1  1
    #> [8,]  1  1  1

If we exclude \(\mathbf{y}_v\)s with identical values, then

    Ymat(3, exclude_identical = TRUE)
    #>      y1 y2 y3
    #> [1,]  1  0  0
    #> [2,]  0  1  0
    #> [3,]  1  1  0
    #> [4,]  0  0  1
    #> [5,]  1  0  1
    #> [6,]  0  1  1

Here is \(\mathbf{Y}\) with corresponding values of AUC:

    cbind(Ymat(3), AUC = apply(Ymat(3), 1, AUC))
    #>      y1 y2 y3 AUC
    #> [1,]  0  0  0 0.0
    #> [2,]  1  0  0 0.0
    #> [3,]  0  1  0 0.5
    #> [4,]  1  1  0 0.0
    #> [5,]  0  0  1 1.0
    #> [6,]  1  0  1 0.5
    #> [7,]  0  1  1 1.0
    #> [8,]  1  1  1 1.0

The values of \(\mathrm{cv}(\widehat{y}_i, y_i)\) that express AUC as a sum of casewise values are solutions of equation (\(\ref{eq:cw}\)), which can be written as solutions of the following system of \(2^{n_v}\) linear simultaneous equations in \(2n_v\) unknowns: \[\begin{equation} \label{eq:lin} \tag{2} (\mathbf{U} -\mathbf{Y}) \mathbf{c}_0 + \mathbf{Y} \mathbf{c}_1 = [\mathbf{U} -\mathbf{Y}, \mathbf{Y}] \begin{bmatrix} \mathbf{c}_0 \\ \mathbf{c}_1 \end{bmatrix} = \mathrm{AUC}(\mathbf{\widehat{Y}},\mathbf{Y}) \end{equation}\] where \(\mathbf{U}_{(2^{n_v} \times n_v)}\) is a matrix of 1s conformable with \(\mathbf{Y}\); \(\mathbf{c}_0 = [\mathrm{cv}(1,0), c(2,0), ..., \mathrm{cv}(n_v,0)]^T\); \(\mathbf{c}_1 = [\mathrm{cv}(1,1), c(2,1), ..., \mathrm{cv}(n_v,1)]^T\); \([\mathbf{U} -\mathbf{Y}, \mathbf{Y}]_{(2^{n_v} \times 2n_v)}\) and \(\begin{bmatrix}\begin{aligned} \mathbf{c}_0 \\ \mathbf{c}_1 \end{aligned} \end{bmatrix}_{(2n_v \times 1)}\) are partitioned matrices; and \(\mathbf{\widehat{Y}}_{(2^{n_v} \times n_v)}\) is a matrix each of whose rows consists of the integers 1 to \(n_v\).

We can test whether equation (\(\ref{eq:lin}\)) has a solution for any given \(n_v\) by trying to solve it as a least-squares problem, considering whether the residuals of the associated linear model are all 0, using the “design matrix” \([\mathbf{U} -\mathbf{Y}, \mathbf{Y}]\) to predict the “outcome” \(\mathrm{AUC}(\mathbf{\widehat{Y}},\mathbf{Y})_{(2^{n_v} \times 1)}\):

    resids <- function(n_v, exclude_identical = FALSE,
                       tol = sqrt(.Machine$double.eps)) {
      Y <- Ymat(n_v, exclude_identical = exclude_identical)
      AUC <- apply(Y, 1, AUC)
      X <- cbind(1-Y, Y)
      opts <- options(warn = -1)
      on.exit(options(opts))
      fit <- lsfit(X, AUC, intercept = FALSE)
      ret <- max(abs(residuals(fit)))
      if(ret < tol){
        ret <- 0
        solution <- coef(fit)
        names(solution) <- paste0("c(", c(1:n_v, 1:n_v), ",",
                                  rep(0:1, each = n_v), ")")
        attr(ret, "solution") <- zapsmall(solution)
      }
      ret
    }

The case \(n_v=3\), excluding identical \(y\)s, has a solution:

    resids(3, exclude_identical = TRUE)
    #> [1] 0
    #> attr(,"solution")
    #> c(1,0) c(2,0) c(3,0) c(1,1) c(2,1) c(3,1)
    #>    1.0    0.0   -0.5    0.5    0.0    0.0

But, if identical \(y\)s are included, the equation is not consistent:

    resids(3, exclude_identical = FALSE)
    #> [1] 0.125

For \(n_v=4\), there are no solutions in either case:

    resids(4, exclude_identical = TRUE)
    #> [1] 0.083333
    resids(4, exclude_identical = FALSE)
    #> [1] 0.25

Consequently, the widely employed AUC measure of fit for binary regression cannot in general be used for a casewise cross-validation criterion.

## References

Fox, J. (2016). Applied regression analysis and generalized linear models (Second edition). Thousand Oaks CA: Sage.
Fox, J., & Weisberg, S. (2019). An R companion to applied regression (Third edition). Thousand Oaks CA: Sage.
Hamner, B., & Frasco, M. (2018). Metrics: Evaluation metrics for machine learning. Retrieved from https://CRAN.R-project.org/package=Metrics
Lüdecke, D., Waggoner, P., & Makowski, D. (2019). insight: A unified interface to access information from model objects in R. Journal of Open Source Software, 4(38), 1412.
Lumley, T., & Miller, A. (2020). leaps: Regression subset selection. Retrieved from https://CRAN.R-project.org/package=leaps
"Receiver operating characteristic". (2023). Receiver operating characteristic—Wikipedia, the free encyclopedia. Retrieved from https://en.wikipedia.org/wiki/Receiver_operating_characteristic
Venables, W. N., & Ripley, B. D. (2002). Modern applied statistics with S (Fourth edition). New York: Springer.

1. This example is somewhat artificial in that `glmmPQL()` has largely been superseded by computationally superior functions, such the `glmer()` function in the lme4 package. There is, however, one situation in which `glmmPQL()` might prove useful: to specify serial dependency in case-level errors within clusters for longitudinal data, which is not currently supported by `glmer()`.↩︎

2. The `regsubsets()` function computes several measures of model predictive performance, including the \(R^2\) and \(R^2\) adjusted for degrees of freedom, the residual sums of squares, Mallows’s \(C_p\), and the BIC. Several of these are suitable for comparing models with differing numbers of coefficients—we use the BIC below—but all necessarily agree when comparing models with the same number of coefficients.↩︎